This is an exercise the book An Introduction to Quantum Stochastic Calculus by K.R. Parthasarathy (Exercise 10.11)
Let $\{\xi_t,t\in T\}$ be a family of observables in $H$ with values in $\Bbb{R}$. Then there exists a measurable space $(\Omega,\mathscr{F})$, and $\Omega$-valued observables $\xi$ in $H$ and a family of real valued measurable functions $\{f_t,t\in T\}$ on $\Omega$ such that $$\int_\Bbb{R}e^{isx}\ d\xi_t(x)=\int_\Omega e^{isf_t(\omega)}\ d\xi(\omega)$$
Here by non-interfering we mean $\xi_t(E)\xi_s(F)=\xi_s(F)\xi_t(E)$ for all $t,s\in T$ and $E,F\subseteq\Bbb{R}$ measurable.
I know the following result by Von-Neumann (which a corollary of Stone's Theorem on the unitary represenations of $\mathbb{R}^k$): Suppose $\xi,\eta$ are $\Bbb{R}^m$ and $\Bbb{R}^n$-valued non-interfering observables in $H$. Then there exists a unique $\Bbb{R}^{m+n}$-valued observable $\zeta$ in $H$ such that $\zeta(E\times F)=\xi(E)\eta(F)$ for all $E\subseteq\Bbb{R}^m$ and $F\subseteq\Bbb{R}^n$.
I can prove that the above result holds for finitely many non-interfering observables. That is in our context, if we have family of non-interfering real valued observables $\{\xi_j\}, j=1,2,\ldots,n$ in $H$. Then there exists $\Bbb{R}^n$ valued observable $\xi$ in $H$ such that $\xi(E_1\times\cdots\times E_n)=\xi_1(E_1)\cdots\xi_n(E_n)$ for all $E_i\subseteq\Bbb{R}$ measurable.
We take $\Omega=\Bbb{R}^n$ and $f_j:\Omega\to\Bbb{R}$ to be the projection onto $j^\text{th}$ component. Then $\xi_j=\xi f_j^{-1}$. Then by the change of variable formula we have the following $$\int_\Bbb{R}e^{isx}\ d\xi_j(x)=\int_\Omega e^{isf_j(\omega)}\ d\xi(\omega)\ \forall j=1,2,\ldots,n$$
But I have to prove for arbitrary collection. There is a hint to use Bochner's Theorem and Zorn's lemma.
Can anyone help me how should I proceed with the help of Bochner's Theorem (to define a positive definite function)? Thanks for your help in advance.